Weighted multidimensional scaling aims at determining object coordinates \(\mathbf {X}=(x_{i\alpha })\in \mathbb {R}^{n\times r}\)  reproducing the dissimilarities \(\mathbf {D}\)  while expressing a maximum amount of dispersion or inertia \(\Delta \)  (REF ) in low dimensions. It is performed by the following weighted generalization of the well-known Torgerson–Gower scaling procedure [1]}: first, define \(\mathbf {\Pi }=\mbox{diag}(\mathbf {f})\) , as well as the weighted centering matrix \(\mathbf {H}=\mathbf {I}_n-\mathbf {1}_n\mathbf {f}^\top \) , which obeys \(\mathbf {H}^2=\mathbf {H}\) . However, \(\mathbf {H}^\top \ne \mathbf {H}\) , unless \(\mathbf {f}\)  is uniform.
